The reducibility theory (consult section REF  for related concepts) has attracted much attention due to its importance in connecting dynamical systems and the spectral theory.
The pioneering work was due to Dinaburg-Sinai [1]}. They first used classical KAM theory to obtain positive measure energies has reducibility for one-dimensional continuous quasi-periodic Schrödinger equation, provided small analytic potential. Eliasson [2]} made a breakthrough by proving the reducibility for almost all energies and the almost reducibility for all energies provided the frequency is Diophantine and the potential is small and analytic, using a crucial resonance-cancelation technique introduced by Moser-Pöschel [3]}.
The non-perturbative reducibility, i.e., the smallness of the perturbations does not depend on the Diophantine constant, was established by Hou-You [4]} (see also Avila-Fayad-Krikorian [5]}).
Recently, Leguil-You-Zhao-Zhou [6]} obtained a strong version of quantitative almost reducibility result, i.e., the strips of width not going to zero. Almost reducibility have many interesting spectral applications, curious readers are invited to consult [7]}.
